Solving Stuff

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Solving is a large part of math.  Before we get deep into solving things, it is important to talk about some the ideas behind it, so that we know what we are doing.  

Variables are the letters stuck in the middle of our math problem.  Why is it a letter?  What is it?  Why should we care?  The variable is a number.  We do not know what number it is, so we just stuck a letter there.  When we figure out what number it is, we don't need to use the letter anymore.  

Solutions are another important thing we need to talk about before explaining how to solve.  A solution is a number that we can put in there where the variable is and get a true math statement.  1=1 is a true math statement, 1 does equal 1 most of the time (if you end up in a graduate math class, they might show that it doesn't anymore).  -3=47 is not true, so that means whatever number you put in for the variable was not a solution, or it means there is no solution to your equation, because nothing will make it true.

Solving means we are going to move stuff around in the equation until we get the variable all alone on one side of the equation.  I like to think of this sort of like a puzzle.  We have to follow the rules, but as long as we follow the rules, we can move stuff all over our equation until we get the variable by itself.  

Rules For Solving:

1. We can do lots of things, as long as we do them to both sides of the equal sign.
   • ​​We can add something to both sides.
     
   • We can subtract something from both sides.
     
   • We can multiply both sides, just make sure you you multiply the whole side.
     
   • We can divide both sides, just make sure you divide the whole side.
     
   • We can square both sides.
     
   • We can take the square root of both sides.
   • We could multiply both sides by something ugly, like (x+3).
     
   • We can take the natural log of both sides, which is sometimes really helpful.
     
   • We can use e on both sides.
2. We want to get the variable alone on one side.  If the variable is on both sides, we need to get it all on one side.  We can move things by doing the opposite of what is already happening in our equation.  So here are the opposites, or inverses.
   
   • ​​Adding and Subtracting
     
   • Multiplying and Dividing
     
   • Squaring and Square Rooting
     
   • Natural Log and e
     
3. Once the variable is on one side, we need to get it alone.  This means we need to undo all of the stuff happening on the side of the equation where our variable is at.  To undo things, we will do the opposite.  In order for us to undo all the stuff happening to our variable, we are going to need to do it in the right order.  Sometimes this means working PEMDAS backwards, but there are some exceptions to this, so it is not a perfect answer.  Solving is sometimes pretty ugly, so you just have to remember that as long as you are following the rules, it is ok, you will eventually get there.  Practice in moving things around in equations is how you eventually become a rock star at solving.  It also helps to see someone who is really good at it do some of the tricks.  The mathematicians have really fancy ways to multiply by one or add zero, both do nothing to change the equation, but if done just right can really change the way it looks.
   
   • ​​Combine everything that you can, x squared goes with x squared, x's can go with x's, numbers can go with numbers.  Combine all of the things on both sides that you can.
     
   • Then get rid of parenthesis by distributing.
     
   • Then combine everything again.
     
   • Then add and subtract stuff to get the variable alone.
     
   • Then multiply and divide stuff to get the variable alone.
     
4. These steps get you through some of the equations, but not all of them.
   

We will start with solving what are called One Step Equations.  This means we only need to do one thing to get the variable alone.  We will build up to solving uglier stuff as we get more practice.  To solve one step equations, look at what is happening to x.  If it is 2x, then x is being multiplied by 2 and in order to undo this, we need to divide both sides of the equation by 2, because we always want to do the opposite.  If we had x/2, then x is being divided by 2, so in order to undo this, we need to multiply both sides by 2.  If it is x-2, then we are subtracting and we need to add 2 to both sides in order to get x alone.

Each of these worksheets has one step solving equations, and they have answer keys so that you can check your work.

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Mr. Lake's Math Site